A lowest order divergence-free finite element on rectangular grids

doi:10.1007/s11464-011-0094-0

Front Math Chin

2011, Vol. 6

Issue (2) : 253-270 https://doi.org/10.1007/s11464-011-0094-0

RESEARCH ARTICLE

A lowest order divergence-free finite element on rectangular grids

Yunqing HUANG¹(

), Shangyou ZHANG²

1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China; 2. Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Download: PDF(873 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

It is shown that the conforming Q2,1;1,2-Q1′ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q2,1;1,2=Q2,1×Q1,2, and Q_2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q1′ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q1′ is the divergence of the discrete velocity space Q_2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.

Keywords Mixed finite element Stokes divergence-free element quadrilateral element rectangular grid

Corresponding Author(s): HUANG Yunqing,Email:huangyq@xtu.edu.cn

Issue Date: 01 April 2011

Cite this article:

Yunqing HUANG,Shangyou ZHANG. A lowest order divergence-free finite element on rectangular grids[J]. Front Math Chin, 2011, 6(2): 253-270.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-011-0094-0
https://academic.hep.com.cn/fmc/EN/Y2011/V6/I2/253

1	Ainsworth M, Coggins P. A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow. IMA J Num Anal , 2002, 22: 307-327 doi: 10.1093/imanum/22.2.307
2	Arnold D N, Qin J. Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky R, Knight D, Richter G, eds. Advances in Computer Methods for Partial Differential Equations VII, IMACS . 1992, 28-34
3	Arnold D N, Scott L R, Vogelius M. Regular inversion of the divergence operator with Dirichlet conditions on a polygon. Ann Sc Norm Super Pisa, C1, Sci, IV Ser , 1988, 15: 169-192
4	Bernardi C, Maday Y. Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math Meth Appl Sci , 1999, 9: 395-414 doi: 10.1142/S0218202599000208
5	Bernardi C, Raugel B. Analysis of some finite elements of the Stokes problem. Math Comp , 1985, 44: 71-79 doi: 10.1090/S0025-5718-1985-0771031-7
6	Brenner S C. An optimal-order multigrid method for P1 nonconforming finite elements. Math Comp , 1989, 52: 1-15
7	Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 1994
8	Brezzi F, Falk F. Stability of higher-order Hood-Taylor methods. SIAM J Numer Anal , 1991, 28(3): 581-590 doi: 10.1137/0728032
9	Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Berlin: Springer, 1991
10	Carrero J, Cockburn B, Sch?tzau D. Hybridized globally divergence-free LDG methods. I. The Stokes problem. Math Comp , 2006, 75: 533-563 doi: 10.1090/S0025-5718-05-01804-1
11	Ciarlet P G. The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978
12	Cockburn C, Li F, Shu C-W. Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J Comput Phys , 2004, 194: 588-610 doi: 10.1016/j.jcp.2003.09.007
13	Fortin M, Glowinski R. Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-value Problems. Amsterdam: North Holland, 1983
14	Karakashian O, Katsaounis T. Numerical simulation of incompressible fluid flow using locally solenoidal elements. Comput Math Appl , 2006, 51: 1551-1570 doi: 10.1016/j.camwa.2005.10.016
15	Li F, Shu C-W. Locally divergence-free discontinuous Galerkin methods for MHD equations. J Sci Comput , 2005, 22-23: 413-442 doi: 10.1007/s10915-004-4146-4
16	Nédélec J-C. Mixed finite elements in ?3. Numer Math , 1980, 35: 315-341 doi: 10.1007/BF01396415
17	Oswald P. Remarks on multilevel bases for divergence-free finite elements. Numerical Algorithms , 2001, 27: 131-152 doi: 10.1023/A:1011893617044
18	Qin J. On the Convergence of Some Low Order Mixed Finite Elements for Incompressible Fluids. Thesis. Pennsylvania State University , 1994
19	Qin J, Zhang S. Stability of the finite elements 9/(4c+1) and 9/5c for stationary Stokes equations. Computers and Structures , 2005, 84: 70-77 doi: 10.1016/j.compstruc.2005.07.002
20	Qin J, Zhang S. Stability and approximability of the P1-P0 element for Stokes equations. Int J Numer Meth Fluids , 2007, 54: 497-515 doi: 10.1002/fld.1407
21	Raviart P A, Girault V. Finite Element Methods for Navier-Stokes Equations. Berlin: Springer, 1986
22	Raviart P A, Thomas J. A mixed finite element method for 2nd order elliptic problems. In: Mathematics Aspects of Finite Element Methods. Lecture Notes in Math, Vol 606 . New York: Springer-Verlag, 1977, 292-315 doi: 10.1007/BFb0064470
23	Scott L R, Vogelius M. Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO, Modelisation Math Anal Numer , 1985, 19: 111-143
24	Scott L R, Vogelius M. Conforming finite element methods for incompressible and nearly incompressible continua. In: Lectures in Applied Mathematics , 22. 1985, 221-244
25	Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp , 1990, 54: 483-493 doi: 10.1090/S0025-5718-1990-1011446-7
26	Scott L R, Zhang S. Multilevel iterated penalty method for mixed elements. In: The Proceedings for the Ninth International Conference on Domain Decomposition Methods, Bergen . 1998, 133-139
27	Stenberg R, Suri M. Mixed hp finite element methods for problems in elasticity and Stokes flow. Numer Math , 1996, 72: 367-389 doi: 10.1007/s002110050174
28	Ye X, Anderson G. The derivation of minimal support basis functions for the discrete divergence operator. J Compu Appl Math , 1995, 61: 105-116 doi: 10.1016/0377-0427(94)00056-7
29	Zhang S. A new family of stable mixed finite elements for 3D Stokes equations. Math Comp , 2005, 74 (250): 543-554 doi: 10.1090/S0025-5718-04-01711-9
30	Zhang S. On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J Comp Math , 2008, 26: 456-470
31	Zhang S. A family of Q_k+1,k × Q_k,k+1 divergence-free finite elements on rectangular grids. SIAM J Num Anal , 2009, 47: 2090-2107 doi: 10.1137/080728949

[1]	Boling GUO, Fengxia LIU. Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations[J]. Front. Math. China, 2019, 14(6): 1133-1161.
[2]	Jishan FAN, Fucai LI. Regularity criteria for Navier-Stokes-Allen-Cahn and related systems[J]. Front. Math. China, 2019, 14(2): 301-314.
[3]	Ruishu WANG, Xiaoshen WANG, Kai ZHANG, Qian ZHOU. Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form[J]. Front. Math. China, 2018, 13(5): 1121-1140.
[4]	Sufang ZHANG,Kaitai LI,Hongen JIA. Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations[J]. Front. Math. China, 2017, 12(2): 481-492.
[5]	Jishan FAN,Fucai LI,Gen NAKAMURA. Two regularity criteria for 3D Navier-Stokes equations in a bounded domain[J]. Front. Math. China, 2017, 12(2): 359-366.
[6]	Yong DING,Xiaochun SUN. Uniqueness of weak solutions for fractional Navier-Stokes equations[J]. Front. Math. China, 2015, 10(1): 33-51.
[7]	Pengzhan HUANG, Yinnian HE, Xinlong FENG. Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations[J]. Front Math Chin, 2013, 8(4): 837-854.
[8]	Yang LIU, Hong LI, Wei GAO, Siriguleng HE, Jinfeng WANG. Splitting positive definite mixed element method for viscoelasticity wave equation[J]. Front Math Chin, 2012, 7(4): 725-742.
[9]	Yanping CHEN, Zuliang LU, Ruyi GUO. Error estimates of triangular mixed finite element methods for quasilinear optimal control problems[J]. Front Math Chin, 2012, 7(3): 397-413.

Viewed

Full text

Abstract

Cited

Shared

Discussed